An analytical model for the vibration of a composite plate containing an embedded periodic shape memory alloy structure

This paper explores the integration of a periodic repeating arrangement of shape memory alloy (SMAs) within a composite plate, with a view to active control of the vibrations of the plate by means of a controllable activation strategy for the SMA elements. The benefits of this configuration are that ‘antagonistic’ operation of SMAs on the plate allows the significantly longer cooling time constant of previously activated elements to be shortened by means of active elements working against them during that phase. This concept dramatically shortens the cooling time constant and brings it into the same order of magnitude of the heating phase. The paper examines the mathematical modelling of such a plate, and offers an approximate analytical solution by means of a hybrid WKB–Galerkin method. The antagonistic operation of the system is represented mathematically by terms in which the stiffness and damping are both time dependent. Therefore the equation of motion contains terms with time variant coefficients and is impossible to solve without recourse to specialised methods. Comparisons with numerical methods are given and it is shown that good similarity can be obtained for judicious choice of practical values for the time variant stiffness and damping functions.